## The Annals of Statistics

- Ann. Statist.
- Volume 20, Number 4 (1992), 2100-2110.

### Invariant Directional Orderings

E. L. Lehmann and J. Rojo

#### Abstract

Statistical concepts of order permeate the theory and practice of statistics. The present paper is concerned with a large class of directional orderings of univariate distributions. (What do we mean by saying that a random variable $Y$ is larger than another random variable $X$?) Attention is restricted to preorders that are invariant under monotone transformations; this includes orderings such as monotone likelihood ratio, hazard ordering, and stochastic ordering. Simple characterizations of these orderings are obtained in terms of a maximal invariant. It is shown how such invariant preorderings can be used to generate concepts of $Y_2$ being further to the right of $X_2$ than $Y_1$ is of $X_1$.

#### Article information

**Source**

Ann. Statist., Volume 20, Number 4 (1992), 2100-2110.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176348905

**Digital Object Identifier**

doi:10.1214/aos/1176348905

**Mathematical Reviews number (MathSciNet)**

MR1193328

**Zentralblatt MATH identifier**

0776.62011

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62A05

Secondary: 62E10: Characterization and structure theory 62B15: Theory of statistical experiments 60E05: Distributions: general theory

**Keywords**

Stochastic ordering monotone likelihood ratio ordering hazard ordering location families distance between distribution functions

#### Citation

Lehmann, E. L.; Rojo, J. Invariant Directional Orderings. Ann. Statist. 20 (1992), no. 4, 2100--2110. doi:10.1214/aos/1176348905. https://projecteuclid.org/euclid.aos/1176348905